Equilibrium Temperatures and Directional Emissivity of Sunlit Airless Surfaces With Applications to the Moon

Solar irradiance dominates the heat flux incident on airless planetary bodies. In thermal equilibrium, surface roughness affects the temperature distribution by changing the incidence angle local to each slope. In order to simulate temperatures and thermal emissions at different phase angles, existing thermophysical models usually employ computationally expensive techniques such as ray tracing. Here we derive the equilibrium surface temperature distribution of sunlit Gaussian rough surfaces, providing an exact solution for the Sun at the zenith and an approximate solution for the general case. We find that although the slope distribution of realistic airless surfaces is often non-Gaussian, their temperature distribution is well modeled assuming a Gaussian slope distribution. We additionally present closed-form expressions that describe the radiation emitted from rough surfaces at different emissions angles and employ them to radiometrically estimate the roughness of the lunar surface using measurements obtained by Lunar Reconnaissance Orbiter (LRO) Diviner. Our model may also be applied to studying the roughness of resolved and unresolved surfaces on other airless planetary bodies.